We investigate the relaxation effects on the dynamics of two-component dilute gas Bose-Einstein condensates (BEC) with relatively different two-body interactions and Josephson couplings between the two components. Three types of relaxation effects, i.e., one- and three-body losses and a pure phase relaxation caused by elastic two-body collision between condensed and noncondensed atoms, are examined on the dynamical behavior of a macroscopic superposition, i.e., Schrödinger cat state, of two states with atom-number differences between the two components, which is known to be created by the time evolution in certain parameter regimes. Although three-body losses show a relatively large suppression of the revival behavior- of Schrödinger cat state and the Pegg-Barnett phase-difference distribution between the two components for a small-size Schrödinger cat state, one- and three-body loss effects are not shown to directly depend on the size of Schrödinger cat state. In contrast, the pure-phase relaxation effects, causing a reduction of phase-difference distribution and then decaying the Schrödinger cat state, significantly increase with the increase of the size of Schrödinger cat state. These features suggest that a detection of damped collapse-revival behavior is highly possible for medium-size Schrödinger cat states in small-size two-component BECs.
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